{"id":177,"date":"2026-03-09T00:30:11","date_gmt":"2026-03-08T16:30:11","guid":{"rendered":"http:\/\/photocaloric.com\/?p=177"},"modified":"2026-03-09T00:32:35","modified_gmt":"2026-03-08T16:32:35","slug":"%e5%88%86%e6%95%b0%e9%87%8f%e5%ad%90%e9%9c%8d%e5%b0%94%e6%95%88%e5%ba%94%e4%ba%8c%ef%bc%9a%e5%8a%b3%e5%a4%ab%e6%9e%97%e6%b3%a2%e5%87%bd%e6%95%b0","status":"publish","type":"post","link":"http:\/\/photocaloric.com\/index.php\/2026\/03\/09\/%e5%88%86%e6%95%b0%e9%87%8f%e5%ad%90%e9%9c%8d%e5%b0%94%e6%95%88%e5%ba%94%e4%ba%8c%ef%bc%9a%e5%8a%b3%e5%a4%ab%e6%9e%97%e6%b3%a2%e5%87%bd%e6%95%b0\/","title":{"rendered":"\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94(\u4e8c\uff1a\u52b3\u592b\u6797\u6ce2\u51fd\u6570)"},"content":{"rendered":"\n

\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94\u4ea7\u751f\u4e8e\u5e26\u6709\u76f8\u4e92\u4f5c\u7528\u7684\u3001\u5782\u76f4\u78c1\u573a\u4e0b\u7684\u4e8c\u7ef4\u7535\u5b50\u4f53\u7cfb<\/strong>\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u7406\u8bba\u4e0a\uff0c\u6211\u4eec\u53ea\u9700\u8981\u5c06\u7535\u5b50\u76f8\u4e92\u4f5c\u7528\u52a0\u5165\u8fdb\u4e4b\u524d\u7684\u7406\u8bba\u4e2d\u5c31\u597d\u4e86\uff1a<\/p>\n\n\n\n

H<\/mi>=<\/mo>(<\/mo>\ud835\udc91<\/mi>+<\/mo>e<\/mi>\ud835\udc68<\/mi>)<\/mo>2<\/mn><\/msup><\/mrow>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>+<\/mo>\u2211<\/mo>i<\/mi>\u2260<\/mo>j<\/mi><\/mrow><\/munder><\/mrow>V<\/mi>(<\/mo>\ud835\udc93<\/mi>i<\/mi><\/msub>,<\/mo>\ud835\udc93<\/mi>j<\/mi><\/msub>)<\/mo><\/mrow>H=\\frac{(\\bm{p}+e\\bm{A})^2}{2m_e}+\\sum_{i\\neq j}V(\\bm{r}_i,\\bm{r}_j)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u8fd9\u662f\u4e00\u4e2a\u591a\u4f53\u54c8\u5bc6\u987f\u91cf\uff0c\u663e\u800c\u6613\u89c1\u662f\u5341\u5206\u590d\u6742\u7684\u3002<\/p>\n\n\n\n

\u4f46\u4ec5\u5bf9\u4e8e2\u7535\u5b50\u7684\u60c5\u51b5\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u731c\u51fa\u672c\u5f81\u6ce2\u51fd\u6570\u3002\u76f4\u63a5\u6839\u636e\u65e0\u76f8\u4e92\u4f5c\u7528\u65f6\u5019\u7684\u6ce2\u51fd\u6570\u5f62\u5f0f\uff1a<\/p>\n\n\n\n

|<\/mi>0<\/mn>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>\u221d<\/mo>z<\/mi>m<\/mi><\/msup>e<\/mi>\u2212<\/mo>|<\/mi>z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/msup><\/mrow>\\ket{0,m}\\propto z^m e^{-\\frac{|z|^2}{4l_B^2}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u76f4\u63a5\u731c\u6d4b2\u7c92\u5b50\u60c5\u51b5\u4e0b\u7684\u672c\u5f81\u6001\u6ce2\u51fd\u6570\u4e3a\uff1a<\/p>\n\n\n\n

|<\/mi>M<\/mi>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>2<\/mn><\/msub>\u221d<\/mo>(<\/mo>z<\/mi>1<\/mn><\/msub>+<\/mo>z<\/mi>2<\/mn><\/msub>)<\/mo>M<\/mi><\/msup>(<\/mo>z<\/mi>1<\/mn><\/msub>\u2212<\/mo>z<\/mi>2<\/mn><\/msub>)<\/mo>m<\/mi><\/msup>e<\/mi>\u2212<\/mo>|<\/mi>z<\/mi>1<\/mn><\/msub>|<\/mi>2<\/mn><\/msup>+<\/mo>|<\/mi>z<\/mi>2<\/mn><\/msub>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/msup><\/mrow>\\ket{M,m}_2\\propto (z_1+z_2)^M(z_1-z_2)^m e^{-\\frac{|z_1|^2+|z_2|^2}{4l_B^2}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u4e2dM\u662f\u4e24\u7c92\u5b50\u7684\u603b\u89d2\u52a8\u91cf\uff0cm\u662f\u76f8\u5bf9\u89d2\u52a8\u91cf\u3002\u9a8c\u8bc1\u5b83\u662f\u4e0a\u8ff0\u54c8\u5bc6\u987f\u91cf\u7684\u65b9\u6cd5\u8f83\u591a\uff0c\u8fd9\u91cc\u9009\u62e9\u6700\u66b4\u529b\u7684\u4e00\u79cd\uff0c\u76f4\u63a5\u5e26\u5165\u859b\u5b9a\u8c14\u65b9\u7a0b\uff1a<\/p>\n\n\n\n

\u222b<\/mo>d<\/mi>\ud835\udc93<\/mi>1<\/mn><\/msub>\u222b<\/mo>d<\/mi>\ud835\udc93<\/mi>2<\/mn><\/msub>\u27e8<\/mo>M<\/mi>\u2032<\/mo><\/msup>,<\/mo>m<\/mi>\u2032<\/mo><\/msup><\/mrow>|<\/mi><\/mpadded>2<\/mn><\/msub>[<\/mo>H<\/mi>01<\/mn><\/msub>+<\/mo>H<\/mi>02<\/mn><\/msub>+<\/mo>V<\/mi>(<\/mo>\ud835\udc93<\/mi>1<\/mn><\/msub>,<\/mo>\ud835\udc93<\/mi>2<\/mn><\/msub>)<\/mo>]<\/mo>|<\/mi>M<\/mi>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>2<\/mn><\/msub><\/mrow>\\int d\\bm{r}_1\\int d\\bm{r}_2\n\\bra{M',m'}_2\n[H_{01}+H_{02}+V(\\bm{r}_1,\\bm{r}_2)]\\ket{M,m}_2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u4e2dH<\/mi>01<\/mn><\/msub>,<\/mo>H<\/mi>02<\/mn><\/msub><\/mrow>H_{01}, H_{02}<\/annotation><\/semantics><\/math>\u662f\u5355\u7c92\u5b50\u54c8\u5bc6\u987f\u91cf\uff0c\u6211\u4eec\u5b9a\u4e49\u76842\u7c92\u5b50\u6ce2\u51fd\u6570\u5f88\u5bb9\u6613\u8bc1\u660e\u662f\u5b83\u4eec\u7684\u672c\u5f81\u6001\uff0c\u56e0\u6b64\u53ef\u4ee5\u5ffd\u7565\u3002\u6211\u4eec\u53ea\u9700\u8981\u8bc1\u660e\u4e09\u9879\u4e2d\u4e0d\u5b58\u5728\u4ea4\u53c9\u9879\uff1a\u27e8<\/mo>M<\/mi>\u2032<\/mo><\/msup>,<\/mo>m<\/mi>\u2032<\/mo><\/msup><\/mrow>|<\/mi><\/mpadded>V<\/mi>|<\/mi>M<\/mi>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>\u221d<\/mo>\u03b4<\/mi>M<\/mi>M<\/mi>\u2032<\/mo><\/msup><\/mrow><\/msub>\u03b4<\/mi>m<\/mi>m<\/mi>\u2032<\/mo><\/msup><\/mrow><\/msub><\/mrow>\\bra{M',m'}V\\ket{M,m}\\propto\\delta_{MM'}\\delta_{mm'}<\/annotation><\/semantics><\/math>\uff0c\u5c31\u80fd\u8bf4\u660e2\u7c92\u5b50\u6ce2\u51fd\u6570\u662f2\u7c92\u5b50\u54c8\u5bc6\u987f\u91cf\u7684\u672c\u5f81\u6001\u3002\u8fd9\u5bf9\u76f8\u4e92\u4f5c\u7528\u52bf\u65bd\u52a0\u4e86\u4e00\u4e2a\u5c0f\u9650\u5236\u2014\u2014\u5b83\u4e00\u5b9a\u662f\u4e2d\u5fc3\u5bf9\u79f0\u7684\uff1a<\/p>\n\n\n\n
V<\/mi>(<\/mo>\ud835\udc93<\/mi>1<\/mn><\/msub>,<\/mo>\ud835\udc93<\/mi>2<\/mn><\/msub>)<\/mo>=<\/mo>V<\/mi>(<\/mo>|<\/mi>\ud835\udc93<\/mi>1<\/mn><\/msub>\u2212<\/mo>\ud835\udc93<\/mi>2<\/mn><\/msub>|<\/mi>)<\/mo><\/mrow>V(\\bm{r}_1,\\bm{r}_2)=V(|\\bm{r}_1-\\bm{r}_2|)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u8fd9\u771f\u662f\u4e00\u4e2a\u5fae\u4e0d\u8db3\u9053\u7684\u8981\u6c42\u3002\u6b64\u5916\uff0c\u5b9a\u4e49Z<\/mi>:<\/mo>=<\/mo>z<\/mi>1<\/mn><\/msub>+<\/mo>z<\/mi>2<\/mn><\/msub><\/mrow>2<\/mn><\/mfrac>,<\/mo>z<\/mi>=<\/mo>z<\/mi>1<\/mn><\/msub>\u2212<\/mo>z<\/mi>2<\/mn><\/msub><\/mrow>Z:=\\frac{z_1+z_2}{2}, z=z_1-z_2<\/annotation><\/semantics><\/math>\uff0c\u4e8e\u662f<\/p>\n\n\n\n
|<\/mi>z<\/mi>1<\/mn><\/msub>|<\/mi>2<\/mn><\/msup>+<\/mo>|<\/mi>z<\/mi>2<\/mn><\/msub>|<\/mi>2<\/mn><\/msup>=<\/mo>2<\/mn>|<\/mi>Z<\/mi>|<\/mi>2<\/mn><\/msup>+<\/mo>1<\/mn>2<\/mn><\/mfrac>|<\/mi>z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>|z_1|^2+|z_2|^2\n=\n2|Z|^2+\\frac{1}{2}|z|^2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
|<\/mi>M<\/mi>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>2<\/mn><\/msub>\u221d<\/mo>(<\/mo>Z<\/mi>)<\/mo>M<\/mi><\/msup>(<\/mo>z<\/mi>)<\/mo>m<\/mi><\/msup>e<\/mi>\u2212<\/mo>2<\/mn>|<\/mi>Z<\/mi>|<\/mi>2<\/mn><\/msup>+<\/mo>1<\/mn>2<\/mn><\/mfrac>|<\/mi>z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/msup><\/mrow>\\ket{M,m}_2\\propto (Z)^M(z)^m e^{-\\frac{2|Z|^2+\\frac{1}{2}|z|^2}{4l_B^2}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
\u222b<\/mo>d<\/mi>\ud835\udc93<\/mi>1<\/mn><\/msub>\u222b<\/mo>d<\/mi>\ud835\udc93<\/mi>2<\/mn><\/msub>\u27e8<\/mo>M<\/mi>\u2032<\/mo><\/msup>,<\/mo>m<\/mi>\u2032<\/mo><\/msup><\/mrow>|<\/mi><\/mpadded>2<\/mn><\/msub>[<\/mo>H<\/mi>01<\/mn><\/msub>+<\/mo>H<\/mi>02<\/mn><\/msub>+<\/mo>V<\/mi>(<\/mo>\ud835\udc93<\/mi>1<\/mn><\/msub>,<\/mo>\ud835\udc93<\/mi>2<\/mn><\/msub>)<\/mo>]<\/mo>|<\/mi>M<\/mi>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>2<\/mn><\/msub><\/mrow>\\int d\\bm{r}_1\\int d\\bm{r}_2\n\\bra{M',m'}_2\n[H_{01}+H_{02}+V(\\bm{r}_1,\\bm{r}_2)]\\ket{M,m}_2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
V<\/mi>(<\/mo>\ud835\udc93<\/mi>1<\/mn><\/msub>,<\/mo>\ud835\udc93<\/mi>2<\/mn><\/msub>)<\/mo>=<\/mo>V<\/mi>(<\/mo>|<\/mi>\ud835\udc93<\/mi>1<\/mn><\/msub>\u2212<\/mo>\ud835\udc93<\/mi>2<\/mn><\/msub>|<\/mi>)<\/mo><\/mrow>V(\\bm{r}_1,\\bm{r}_2)=V(|\\bm{r}_1-\\bm{r}_2|)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u4e8e\u662f\u539f\u5f0f\u5bb9\u6613\u5f97\u5230<\/p>\n\n\n\n

\u222b<\/mo>d<\/mi>\ud835\udc93<\/mi>1<\/mn><\/msub>\u222b<\/mo>d<\/mi>\ud835\udc93<\/mi>2<\/mn><\/msub>\u27e8<\/mo>M<\/mi>\u2032<\/mo><\/msup>,<\/mo>m<\/mi>\u2032<\/mo><\/msup><\/mrow>|<\/mi><\/mpadded>2<\/mn><\/msub>[<\/mo>V<\/mi>(<\/mo>\ud835\udc93<\/mi>1<\/mn><\/msub>,<\/mo>\ud835\udc93<\/mi>2<\/mn><\/msub>)<\/mo>]<\/mo>|<\/mi>M<\/mi>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>2<\/mn><\/msub>=<\/mo>(<\/mo>\u222b<\/mo>(<\/mo>Z<\/mi>M<\/mi>\u2032<\/mo><\/msup><\/msup>)<\/mo><\/mrow>\u2217<\/mo><\/msup>Z<\/mi>M<\/mi><\/msup>e<\/mi>\u2212<\/mo>|<\/mi>Z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mfrac><\/mrow><\/msup>d<\/mi>2<\/mn><\/msup>Z<\/mi>)<\/mo><\/mrow>(<\/mo>\u222b<\/mo>(<\/mo>z<\/mi>m<\/mi>\u2032<\/mo><\/msup><\/msup>)<\/mo><\/mrow>\u2217<\/mo><\/msup>z<\/mi>m<\/mi><\/msup>e<\/mi>\u2212<\/mo>|<\/mi>Z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/msup>V<\/mi>(<\/mo>|<\/mi>z<\/mi>|<\/mi>)<\/mo>d<\/mi>2<\/mn><\/msup>z<\/mi>)<\/mo><\/mrow><\/mrow>\\int d\\bm{r}_1\\int d\\bm{r}_2\n\\bra{M',m'}_2\n[V(\\bm{r}_1,\\bm{r}_2)]\\ket{M,m}_2\n=\n\\left(\\int \\left(Z^{M'}\\right)^*Z^M e^{-\\frac{|Z|^2}{l_B^2}} d^2Z \\right)\n\\left(\\int \\left(z^{m'}\\right)^*z^m e^{-\\frac{|Z|^2}{4l_B^2}} V(|z|) d^2z \\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5c06\u590d\u6570\u753b\u4e3a\u6a21\u548c\u5e45\u89d2\u7684\u5f62\u5f0fZ<\/mi>=<\/mo>R<\/mi>e<\/mi>i<\/mi>\u03b8<\/mi><\/mrow><\/msup>,<\/mo>z<\/mi>=<\/mo>r<\/mi>e<\/mi>\u03c6<\/mi><\/msup><\/mrow>Z=Re^{i\\theta}, z=re^{\\varphi}<\/annotation><\/semantics><\/math>\u5f62\u5f0f\uff0c\u5f88\u5bb9\u6613\u770b\u51fa\u5176\u4e2d\u5305\u542b\u7684\u9879<\/p>\n\n\n\n
\u222b<\/mo>d<\/mi>\u03b8<\/mi>e<\/mi>i<\/mi>(<\/mo>M<\/mi>\u2032<\/mo><\/msup>\u2212<\/mo>M<\/mi>)<\/mo>\u03b8<\/mi><\/mrow><\/msup>=<\/mo>2<\/mn>\u03c0<\/mi>\u03b4<\/mi>M<\/mi>M<\/mi>\u2032<\/mo><\/msup><\/mrow><\/msub><\/mrow>\\int d\\theta e^{i(M'-M)\\theta}\n=\n2\\pi\\delta_{MM'}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
\u222b<\/mo>d<\/mi>\u03b8<\/mi>e<\/mi>i<\/mi>(<\/mo>M<\/mi>\u2032<\/mo><\/msup>\u2212<\/mo>M<\/mi>)<\/mo>\u03b8<\/mi><\/mrow><\/msup>=<\/mo>2<\/mn>\u03c0<\/mi>\u03b4<\/mi>M<\/mi>M<\/mi>\u2032<\/mo><\/msup><\/mrow><\/msub><\/mrow>\\int d\\theta e^{i(M'-M)\\theta}\n=\n2\\pi\\delta_{MM'}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u4e8e\u662f\u6211\u4eec\u5c31\u8bc1\u660e\u4e86\u76f8\u4e92\u4f5c\u7528\u9879\u5e76\u4e0d\u4f1a\u5728\u4e0d\u540c\u53c2\u6570\u7684\u6ce2\u51fd\u6570\u4e2d\u5f15\u5165\u4ea4\u53c9\u9879\u3002\u8fd9\u610f\u5473\u7740\u4ec0\u4e48\uff1f\u610f\u5473\u7740\u6211\u4eec\u731c\u6d4b\u7684\u8fd9\u4e2a\u6ce2\u51fd\u6570\u5c31\u662f\u5b58\u5728\u76f8\u4e92\u4f5c\u7528\u60c5\u51b5\u4e0b\u76842\u7535\u5b50\u4f53\u7cfb\u7684\u672c\u5f81\u6001\uff01\u6ce8\u610f\u4ee5\u4e0a\u63a8\u5bfc\u6211\u4eec\u53ea\u8bc1\u660e\u4e86\uff1a\u5f53\u4f53\u7cfb\u4e2d\u4ec5\u5b58\u5728\u6700\u4f4e\u6717\u9053\u80fd\u7ea7\u4e2d\u7684\u7535\u5b50\u65f6\uff08\u5373n=0\uff09\uff0c\u8be5\u4e8c\u4f53\u6ce2\u51fd\u6570\u662f\u54c8\u5bc6\u987f\u91cf\u7684\u672c\u5f81\u6001\uff0c\u5f53\u7535\u5b50\u586b\u5145\u5230\u66f4\u9ad8\u6717\u9053\u80fd\u7ea7\u540e\uff0c\u8fd9\u4e2a\u65ad\u8a00\u5e76\u4e0d\u663e\u7136\u3002<\/p>\n\n\n\n

\u7136\u800c\uff0c\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94\u9762\u5bf9\u7684\u662f\u4e00\u4e2a\u7535\u5b50\u6d53\u5ea6\u4e3an<\/mi>e<\/mi><\/msub>\u2248<\/mo>10<\/mn>10<\/mn>\u2212<\/mo>12<\/mn><\/mrow><\/msup>c<\/mi>m<\/mi>\u2212<\/mo>12<\/mn><\/mrow><\/msup><\/mrow>n_e\\approx 10^{10-12} cm^{-12}<\/annotation><\/semantics><\/math>\u7684\u4f53\u7cfb\uff0c\u5176\u4e2d\u5305\u542b\u5b8f\u89c2\u6570\u91cf\u7684\u7535\u5b50\uff0c\u8fd9\u662f\u4e00\u4e2a\u5178\u578b\u7684\u591a\u4f53\u91cf\u5b50\u7cfb\u7edf\u3002\u9762\u5bf9\u8fd9\u6837\u4e00\u4e2a\u591a\u4f53\u91cf\u5b50\u4f53\u7cfb\uff0c\u6b63\u5e38\u4eba\u90fd\u4e0d\u4f1a\u60f3\u7740\u8bf4\u662f\u5148\u628a\u6ce2\u51fd\u6570\u5199\u51fa\u6765\uff01\u4e00\u822c\u7684\u505a\u6cd5\u662f\u4ece\u54c8\u5bc6\u987f\u91cf\u51fa\u53d1\uff0c\u901a\u8fc7\u5fae\u6270\u8bba\u3001\u5e73\u5747\u573a\u6216\u6709\u6548\u573a\u8bba\u7b49\u65b9\u6cd5\u6c42\u51fa\u80fd\u8c31\u3001\u6ce2\u51fd\u6570\u7b49\u6027\u8d28\u3002\u7136\u800c\uff0c\u52b3\u592b\u6797\uff08Laughlin\uff09\u5219\u50cf\u662f\u6536\u5230\u4e86\u795e\u8c15\u4e00\u6837\uff0c\u8d8a\u8fc7\u4e86\u8fc7\u7a0b\u3001\u76f4\u63a5\u7ed9\u51fa\u4e86\u7ed3\u679c\uff08\u8fd9\u4e2a\u4f53\u7cfb\u7684\u6ce2\u51fd\u6570\uff09\u2014\u2014Laughlin \u6ce2\u51fd\u6570\uff1a<\/p>\n\n\n\n
\u03a8<\/mi><\/mrow>m<\/mi><\/msub>(<\/mo>{<\/mo>z<\/mi>}<\/mo>)<\/mo>\u221d<\/mo>(<\/mo>\u220f<\/mo>i<\/mi><<\/mo>j<\/mi><\/mrow><\/munder><\/mrow>(<\/mo>z<\/mi>i<\/mi><\/msub>\u2212<\/mo>z<\/mi>j<\/mi><\/msub>)<\/mo>m<\/mi><\/msup>)<\/mo><\/mrow><\/mspace>exp<\/mi>\u2061<\/mo><\/mspace><\/mrow>(<\/mo>\u2212<\/mo>\u2211<\/mo>i<\/mi><\/munder><\/mrow>|<\/mi>z<\/mi>i<\/mi><\/msub>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac>)<\/mo><\/mrow><\/mrow>\\Psi_m(\\{z\\})\\propto \\left(\\prod_{i<j} (z_i-z_j)^m\\right)\n \\exp{\\left( -\\sum_i\\frac{|z_i|^2}{4l_B^2} \\right)}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u4e2dm\u662f\u4e00\u4e2a\u5947\u6570\u3002\u8fd9\u4e2a\u6ce2\u51fd\u6570\u5f88\u5bb9\u6613\u770b\u51fa\u590d\u5408\u8d39\u7c73\u5b50\u7684\u6ce1\u5229\u4e0d\u76f8\u5bb9\u6027\u8d28\u2014\u2014\u4ea4\u6362\u4e24\u4e2a\u7c92\u5b50\u7684\u4f4d\u7f6e\u53d8\u53f7\uff0c\u4ee5\u53ca\u4e24\u7c92\u5b50\u4e0d\u80fd\u5904\u4e8e\u540c\u4e00\u4e2a\u4f4d\u7f6e\uff08\u5426\u5219\u6ce2\u51fd\u6570\u4e3a0\uff09\u3002\u6b64\u5916\uff0c\u7535\u5b50\u4e5f\u4e0d\u80fd\u76f8\u9694\u65e0\u7a77\u8fdc\uff0c\u5426\u5219\u6ce2\u51fd\u6570\u4e2d\u7684\u6307\u6570\u9879\u4e5f\u8d8b\u4e8e0\u3002\u5f88\u5bb9\u6613\u770b\u51fa\u8fd9\u4e2a\u6ce2\u51fd\u6570\u548c2\u7535\u5b50\u4f53\u7cfb\u7684\u6ce2\u51fd\u6570\u5f88\u50cf\u3002<\/p>\n\n\n\n

\u65e0\u6cd5\u8bc1\u660e\u8fd9\u4e2a\u6ce2\u51fd\u6570\u7684\u786e\u662f\u591a\u4f53\u54c8\u5bc6\u987f\u91cf\u7684\u672c\u5f81\u6001\uff0c\u4f46\u6211\u4eec\u53ef\u4ee5\u65b9\u4fbf\u9a8c\u8bc1\u5b83\u7684\u786e\u80fd\u591f\u89e3\u91ca\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94\u7684\u8bb8\u591a\u6027\u8d28\u3002\u9996\u5148\uff0c\u6211\u4eec\u5c06\u8bf4\u660e\u5b83\u7684\u786e\u63cf\u8ff0\u4e86\u03bd<\/mi>=<\/mo>1<\/mn>m<\/mi><\/mfrac><\/mrow>\\nu=\\frac{1}{m}<\/annotation><\/semantics><\/math>\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6001\u3002\u56de\u987e\u586b\u5145\u6570\u7684\u8ba1\u7b97\u65b9\u6cd5\uff0c\u662f\u8bf4\uff0c\u5047\u8bbe\u4e8c\u7ef4\u4f53\u7cfb\u7684\u603b\u9762\u79ef\u4e3aS\uff0c\u90a3\u4e48\u90a3\u4e48\u603b\u7684\u78c1\u901a\u662fSB\uff0c\u5355\u4e2a\u6717\u9053\u80fd\u7ea7\uff08\u4e0d\u8003\u8651\u81ea\u65cb\uff09\u7684\u7b80\u5e76\u5ea6\u662f\u5176\u4e2d\u7684\u91cf\u5b50\u78c1\u901a\u6570\uff1a<\/p>\n\n\n\n
N<\/mi>L<\/mi><\/msub>=<\/mo>S<\/mi>B<\/mi><\/mrow>\u03d5<\/mi>B<\/mi><\/msub><\/mfrac><\/mrow>N_L=\\frac{SB}{\\phi_B}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5047\u8bbe\u6bcf\u4e2a\u7535\u5b50\u6ce2\u51fd\u6570\u5360\u636e\u7684\u9762\u79ef\u4e3aA\uff0c\u90a3\u4e48\u603b\u7535\u5b50\u6570\u4e3aS\/A\u3002\u6240\u6709\u7535\u5b50\u6240\u5360\u636e\u7684\u6717\u9053\u80fd\u7ea7\u6570\u91cf\uff0c\u5373\u586b\u5145\u6570\uff1a<\/p>\n\n\n\n

\u03bd<\/mi>=<\/mo>N<\/mi>L<\/mi><\/msub>S<\/mi>\/<\/mi>A<\/mi><\/mrow><\/mfrac>=<\/mo>A<\/mi>B<\/mi><\/mrow>\u03d5<\/mi>B<\/mi><\/msub><\/mfrac>=<\/mo>A<\/mi>2<\/mn>\u03c0<\/mi>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow>\\nu=\\frac{N_L}{S\/A}=\\frac{AB}{\\phi_B}=\\frac{A}{2\\pi l_B^2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Laughlin\u6ce2\u51fd\u6570\u7684\u9762\u79ef\u2014\u2014\u6839\u636e\u5355\u7c92\u5b50\u60c5\u51b5\u65f6\u5f97\u5230\u7684\u7ed3\u8bba\uff0c\u548c\u5176\u89d2\u52a8\u91cf\u6709\u5173\uff1aR<\/mi>=<\/mo>2<\/mn>m<\/mi><\/mrow><\/msqrt>l<\/mi>B<\/mi><\/msub>,<\/mo>A<\/mi>=<\/mo>\u03c0<\/mi>R<\/mi>2<\/mn><\/msup>=<\/mo>2<\/mn>\u03c0<\/mi>m<\/mi>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow>R=\\sqrt{2m}l_B, A=\\pi R^2=2\\pi ml_B^2<\/annotation><\/semantics><\/math>\u3002Laughlin\u6ce2\u51fd\u6570\u4e2d\uff0c\u5bf9\u4e8e\u5355\u4e2a\u7c92\u5b50\uff0c\u4f8b\u5982z<\/mi>1<\/mn><\/msub>z_1<\/annotation><\/semantics><\/math>\uff0c\u5bb9\u6613\u770b\u51fa\u5176\u6700\u9ad8\u6b21\u5e42\u4e3am<\/mi>(<\/mo>N<\/mi>\u2212<\/mo>1<\/mn>)<\/mo><\/mrow>m(N-1)<\/annotation><\/semantics><\/math>\u3002\u4f46\u662fLaughlin\u6ce2\u51fd\u6570\u5305\u542b\u4f53\u7cfb\u4e2d\u6240\u6709\u7535\u5b50\uff0c\u56e0\u6b64\u5e73\u5747\u5355\u7c92\u5b50\u5360\u636e\u7684\u9762\u79ef\u5927\u81f4\u4e3a<\/p>\n\n\n\n
A<\/mi>=<\/mo>2<\/mn>\u03c0<\/mi>m<\/mi>(<\/mo>N<\/mi>\u2212<\/mo>1<\/mn>)<\/mo>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow>N<\/mi><\/mfrac>\u2248<\/mo>2<\/mn>\u03c0<\/mi>m<\/mi>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow>A=\\frac{2\\pi m(N-1)l_B^2}{N} \\approx 2\\pi ml_B^2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u4e8e\u662f\u586b\u5145\u6570\u4e3a<\/p>\n\n\n\n

\u03bd<\/mi>\u2248<\/mo>1<\/mn>m<\/mi><\/mfrac><\/mrow>\\nu\\approx\\frac{1}{m}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u7531\u6ce2\u51fd\u6570\u7684\u53cd\u5bf9\u79f0\u8981\u6c42\uff0cm\u4e3a\u5947\u6570\uff0c\u56e0\u6b64laughlin\u6ce2\u51fd\u6570\u5b9e\u9645\u4e0a\u53ef\u4ee5\u63cf\u8ff0\u5947\u6570\u5206\u6bcd\u3001\u5206\u5b50\u4e3a1\u7684\u6001\u3002<\/p>\n\n\n\n

\u7136\u800c\uff0c\u5b9e\u9645\u4e0a\uff0c\u6570\u503c\u5206\u6790\u8868\u660e\uff0claughlin\u6ce2\u51fd\u6570\u4ec5\u5728\u5c11\u91cf\u7535\u5b50\uff08\u4f8b\u5982\u51e0\u5341\u4e2a\uff09\u65f6\uff0c\u51e0\u4e4e\u662f\u8be5\u4f53\u7cfb\u7684\u7cbe\u786e\u7684\u672c\u5f81\u6001\uff0c\u800c\u5f53\u7535\u5b50\u6570\u975e\u5e38\u5927\u65f6\uff0c\u4e0e\u6570\u503c\u89e3\u51e0\u4e4e\u6ca1\u6709\u91cd\u53e0\uff0c\u8fd9\u8bf4\u660e\u5b83\u4e0d\u662f\u771f\u5b9e\u5206\u6570\u91cf\u5b50\u970d\u5c14\u4f53\u7cfb\u7684\u672c\u5f81\u6ce2\u51fd\u6570\u3002\u7136\u800c\uff0c\u5b83\u7684\u786e\u80fd\u591f\u5f88\u597d\u5730\u63cf\u8ff0\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94\u7684\u5927\u591a\u6570\u6027\u8d28\uff0c\u56e0\u6b64\u6211\u4eec\u8ba4\u4e3a\u5b83\u4e0e\u771f\u5b9e\u7684\u6ce2\u51fd\u6570\u5341\u5206\u7c7b\u4f3c\uff0c\u5373\u201c\u5904\u4e8e\u540c\u4e00\u4e2a\u666e\u9002\u7c7b\u201d\u3002\uff08\u5927\u6982\u662f\u6307\uff0c\u5177\u6709\u540c\u6837\u7684\u5206\u6570\u6fc0\u53d1\uff0c\u4ee5\u53ca\u540c\u6837\u7684\u62d3\u6251\u5e8f\u3002\uff09<\/p>\n\n\n\n

\u5b9e\u9645\u4e0a\uff0c\u989d\uff0c\u5728\u03bd<\/mi>=<\/mo>1<\/mn><\/mrow>\\nu=1<\/annotation><\/semantics><\/math>\u65f6\uff0c\u5373\u6700\u4f4e\u6717\u9053\u80fd\u7ea7\u5b8c\u5168\u586b\u6ee1\u65f6\uff08\u4e0d\u8003\u8651\u81ea\u65cb\uff09\uff0claughlin\u6ce2\u51fd\u6570\u5e94\u5f53\u4e5f\u662f\u65e0\u76f8\u4e92\u4f5c\u7528\u4e8c\u7ef4\u7535\u5b50\u4f53\u7cfb\u7684\u672c\u5f81\u51fd\u6570\uff0c\u80fd\u591f\u63cf\u8ff0\u6574\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94\u3002\u6211\u4eec\u5c06\u4ece\u96f6\u5f00\u59cb\u6784\u5efa\u6ee1\u586b\u5145\u7684\u6700\u4f4e\u6717\u9053\u80fd\u7ea7\u7684\u6ce2\u51fd\u6570\uff0c\u7136\u540e\u548cm=1\u7684laughlin\u6ce2\u51fd\u6570\u6bd4\u8f83\uff0c\u8bf4\u660e\u5b83\u4eec\u5b8c\u5168\u76f8\u540c\u3002<\/p>\n\n\n\n

\uff0c\uff0c\uff0c<\/p>\n\n\n\n

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